From a set of N financial products (N>1), an infinite number of portfolios are available for investment. For purposes of this application, the term “financial product” is broadly defined as a legal representation of the right (often denoted as a claim or security) to provide or receive prospective future benefits under certain stated conditions. For example, domestic or foreign common stock, domestic or foreign bonds, real estate, cash equivalents, mutual funds, exchange traded funds (ETFs) and other securities or portfolios of securities and the like are contemplated by the term “financial product.”
At any rate, existing computer financial analysis systems (also referred to as “portfolio optimizers”) purport to help individuals select portfolios to meet their needs. These systems typically implement mathematical models based upon standard optimization techniques involving mean-variance optimization theory. According to the mean-variance approach to portfolio selection, an optimal portfolio of financial products may be identified with reference to an investor's preference for various combinations of risk and return and the set of efficient portfolios (also referred to as the efficient set or the efficient frontier). FIG. 1 illustrates a feasible set of portfolios that represents all the portfolios that may be formed from a particular set of financial products. The arc AC represents an efficient set of portfolios that each provides the highest expected return for a given level of risk. A portfolio's risk is typically measured by the standard deviation of returns.
In general, the process of portfolio optimization involves determining a portfolio of financial products that maximize the utility function of an investor. Typically portfolio optimization processing assumes that users have a utility function reasonably approximated by a quadratic utility function (i.e. mean-variance utility), namely, that people like having more wealth on average and dislike volatility of wealth. Based on this assumption and given a user's risk tolerance, an optimized portfolio that is mean-variance efficient is calculated from the set of financial products available to the user.
The problem of determining an optimized portfolio from a set of N financial products may be expressed as a series of one or more Quadratic programming (QP) problems. QP is a technique for optimizing (minimizing or maximizing) a quadratic function of decision variables subject to linear equality and or inequality constraints on those decision variables. One particular type of QP technique, referred to as the “active set” method, has been utilized in financial analysis products by Financial Engines, Inc. (FEI). The “active set” method is described in Gill, Murray, and Wright, “Practical Optimization”, Academic Press, 1981, Chapter 5 which is hereby incorporated by reference. Another frequently used method is the critical-line algorithm, which is described in Markowitz, Portfolio Selection, 2nd edition, Blackwell, Cambridge, Mass. 1991, which is hereby incorporated by reference. Additionally, if the optimization problem is appropriately constrained, then it may be solved using the gradient method of QP as described in Sharpe, “An Algorithm for Portfolio Improvement,” Advances in Mathematical programming and Financial Planning, JAI Press, Inc., 1987, which is hereby incorporated by reference.
Current QP optimization programs do not have the capability of factoring the payment of a load either at the front end or back end of the purchase of a financial product since the inclusion of a load significantly changes the form of the function being optimized, i.e. the objective function is no longer quadratic. In short, using prior art optimization QP techniques, it is not possible to easily and efficiently determine an optimum portfolio for an investor by reallocating his current portfolio when his portfolio already includes a portion of loaded financial products.